If the ratio of two numbers is 17 : 6 and the product of their LCM and their HCF is 102, then the sum of the reciprocals of the LCM and the HCF is:​

Correct option is C

Given:

Ratio of two numbers = 17 : 6 (coprime).

​$$\text{LCM} \times \text{HCF} = 102$$​​

Formula Used:

If numbers are in ratio m : n with gcd(m ,n) = 1, then numbers = mH, nH where H = HCF.

LCM(mH , nH) = mnH.

For any two numbers:

LCM $$\times$$​ HCF = product of numbers

Solution:
Let the numbers be (17H) and (6H).
​$$\text{LCM}=17\times 6 \times H = 102$$​H,   HCF = H
Given $$\text{LCM}\times \text{HCF}=$$​102:
(102H)$$\times$$​H = 102

10$$2H^2=102$$​​

H = 1
So HCF = 1, LCM = 102.
Sum of reciprocals:

​$$\frac{1}{\text{LCM}}+\frac{1}{\text{HCF}}=\frac{1}{102}+1=\frac{103}{102}.​​$$​​